Module 5: Doing math with R. Part 1
A=matrix(1:100, nrow=10)
B=matrix(1:1000, nrow=10)
And told to find an inverse of a matrix and determinant of a matrix using these values.
The goal here is starting with these values, we will manipulate these in order for us to attain an inverse matrix and a determinant.
Github link : https://github.com/Ant-nguyen/Intro_r_2021/blob/main/Module%205.R
Let us begin
First we must create matrices A and B in R. To do this we simply assign the matrices to the corresponding variables:
A <- matrix(1:100, nrow=10)
B <- matrix(1:1000, nrow=10)
We get the following matrices:
A <- matrix(1:100, nrow = 10)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1 11 21 31 41 51 61 71 81 91
[2,] 2 12 22 32 42 52 62 72 82 92
[3,] 3 13 23 33 43 53 63 73 83 93
[4,] 4 14 24 34 44 54 64 74 84 94
[5,] 5 15 25 35 45 55 65 75 85 95
[6,] 6 16 26 36 46 56 66 76 86 96
[7,] 7 17 27 37 47 57 67 77 87 97
[8,] 8 18 28 38 48 58 68 78 88 98
[9,] 9 19 29 39 49 59 69 79 89 99
[10,] 10 20 30 40 50 60 70 80 90 100
B <- matrix(1:1000,nrow = 10)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15] [,16]
[1,] 1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151
[2,] 2 12 22 32 42 52 62 72 82 92 102 112 122 132 142 152
[3,] 3 13 23 33 43 53 63 73 83 93 103 113 123 133 143 153
[4,] 4 14 24 34 44 54 64 74 84 94 104 114 124 134 144 154
[5,] 5 15 25 35 45 55 65 75 85 95 105 115 125 135 145 155
[6,] 6 16 26 36 46 56 66 76 86 96 106 116 126 136 146 156
[7,] 7 17 27 37 47 57 67 77 87 97 107 117 127 137 147 157
[8,] 8 18 28 38 48 58 68 78 88 98 108 118 128 138 148 158
[9,] 9 19 29 39 49 59 69 79 89 99 109 119 129 139 149 159
[10,] 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
...
...
...
[,77] [,78] [,79] [,80] [,81] [,82] [,83] [,84] [,85] [,86] [,87] [,88] [,89] [,90] [,91]
[1,] 761 771 781 791 801 811 821 831 841 851 861 871 881 891 901
[2,] 762 772 782 792 802 812 822 832 842 852 862 872 882 892 902
[3,] 763 773 783 793 803 813 823 833 843 853 863 873 883 893 903
[4,] 764 774 784 794 804 814 824 834 844 854 864 874 884 894 904
[5,] 765 775 785 795 805 815 825 835 845 855 865 875 885 895 905
[6,] 766 776 786 796 806 816 826 836 846 856 866 876 886 896 906
[7,] 767 777 787 797 807 817 827 837 847 857 867 877 887 897 907
[8,] 768 778 788 798 808 818 828 838 848 858 868 878 888 898 908
[9,] 769 779 789 799 809 819 829 839 849 859 869 879 889 899 909
[10,] 770 780 790 800 810 820 830 840 850 860 870 880 890 900 910
[,92] [,93] [,94] [,95] [,96] [,97] [,98] [,99] [,100]
[1,] 911 921 931 941 951 961 971 981 991
[2,] 912 922 932 942 952 962 972 982 992
[3,] 913 923 933 943 953 963 973 983 993
[4,] 914 924 934 944 954 964 974 984 994
[5,] 915 925 935 945 955 965 975 985 995
[6,] 916 926 936 946 956 966 976 986 996
[7,] 917 927 937 947 957 967 977 987 997
[8,] 918 928 938 948 958 968 978 988 998
[9,] 919 929 939 949 959 969 979 989 999
[10,] 920 930 940 950 960 970 980 990 1000
One condition to obtaining a determinant or inverse of a matrix is the matrix must be a square matrix. What this means is that the rows and columns of the matrix must be the same. Since Matrix A is already square(10x10), we will begin first with Matrix A.
Matrix A
We will be using the following functions in R to obtain the determinant and inverse matrix :
det() # will return the determinant of a matrix
solve()# will return an inverse matrix if an invertible matrix is inputted
In order for a matrix to have an inverse matrix it must have a determinant , so we will first find the determinant of Matrix A with :
>det(A)
[1] 0
Uh oh, a determinant value of 0 indicates that the matrix does not have an inverse matrix. We can test this with solve(A):
>solve(A)
Error in solve.default(A) :
Lapack routine dgesv: system is exactly singular: U[6,6] = 0
As expected, R is telling us the matrix is singular and can not produce an inverse matrix.
So now what?!
Well one thing we can do to our current matrix is transpose it. What this essentially does is flip the rows and columns of the matrix (i.e: element in [1,2] would be swapped with element in [2,1]), hopefully this gives us a matrix with a determinant and therefore an inverse.
To do this in R, we simply use the t() function.
>a <-t(A)# we assign the transpose of matrix A to variable a
This is our result:
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1 2 3 4 5 6 7 8 9 10
[2,] 11 12 13 14 15 16 17 18 19 20
[3,] 21 22 23 24 25 26 27 28 29 30
[4,] 31 32 33 34 35 36 37 38 39 40
[5,] 41 42 43 44 45 46 47 48 49 50
[6,] 51 52 53 54 55 56 57 58 59 60
[7,] 61 62 63 64 65 66 67 68 69 70
[8,] 71 72 73 74 75 76 77 78 79 80
[9,] 81 82 83 84 85 86 87 88 89 90
[10,] 91 92 93 94 95 96 97 98 99 100
Now we can test to see if this matrix has a determinant:
>det(a)
[1] 3.49659e-113
Since matrix a has a determinant it technically has an inverse matrix,
but when we try solve(a), we get the following error:
Error in solve.default(a) :
system is computationally singular: reciprocal condition number = 2.46047e-19
Well from looking online and searching the RStudios help menu(See reference below if interested). The error is essentially telling us that by default solve() has a reciprocal condition number limit, numbers lower than that limit R considers “computationally singular”. Luckily solve() has an argument called tol which allows us to bypass this limit. R tells us what the reciprocal condition number of matrix a is (2.46047e-19) and knowing this we can make the limit lower than this value, such as 1e-20.
> solve(a,tol= 1e-20)
Our result:
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] -5.634484e+14 5.802766e+14 -8.248661e+13 -1.235905e+11 5.587555e+13 1.652204e+14
[2,] -7.327203e+14 8.295997e+14 -4.955475e+13 2.505479e+12 8.134810e+13 9.409855e+13
[3,] 1.864798e+15 -1.926228e+15 2.100562e+14 -4.309536e+12 -3.315649e+14 -4.114933e+14
[4,] 1.566059e+15 -1.700085e+15 1.035485e+14 -2.868815e+12 -1.650109e+14 -2.013593e+14
[5,] 4.431985e+15 -6.553898e+15 3.332248e+15 -8.857555e+12 -3.354985e+14 -2.143182e+15
[6,] -7.114063e+15 9.350894e+15 -3.598828e+15 1.572532e+13 7.501241e+14 2.662606e+15
[7,] -3.110972e+14 3.223602e+14 -1.541846e+14 1.406863e+14 1.001476e+14 2.699667e+13
[8,] 5.753854e+14 -6.767767e+14 2.734311e+14 -1.413169e+14 1.113297e+13 -2.642284e+14
[9,] -5.438296e+15 7.577206e+15 -3.303103e+15 -1.295905e+14 5.116423e+14 2.361788e+15
[10,] 5.721398e+15 -7.803349e+15 3.268873e+15 1.281498e+14 -6.781964e+14 -2.290447e+15
[,7] [,8] [,9] [,10]
[1,] 1.960053e+13 -1.082916e+12 1.007563e+13 -1.839072e+14
[2,] -1.110828e+14 -5.184998e+13 -5.665069e+13 -5.693305e+12
[3,] 7.002996e+13 1.203892e+14 3.503181e+13 3.732902e+14
[4,] 8.180262e+13 1.062553e+14 -2.890647e+13 2.405648e+14
[5,] -3.486459e+14 3.392499e+14 -1.116162e+14 1.398215e+15
[6,] 3.074136e+14 -5.844309e+14 1.624414e+14 -1.951882e+15
[7,] -5.194450e+13 -2.014751e+13 4.357206e+13 -9.638895e+13
[8,] 4.459921e+13 4.229854e+13 -4.751712e+13 1.829919e+14
[9,] 3.114425e+14 -4.383910e+14 1.633743e+14 -1.616072e+15
[10,] -3.232152e+14 4.877093e+14 -1.698048e+14 1.658882e+15It may not be pretty but at least we were able to get an inverse matrix using Matrix A.
Matrix B
Now for Matrix B, the first thing that stands out is that Matrix B is not a square matrix. It is 10x100.
One way to fix this issue is to matrix multiple Matrix B with a 100x10, the product of this matrix will be a 10x10 matrix. One 100x10 matrix that come to mind is the transpose of Matrix B.
Again we do this by using the t() function:
> b <- t(B)
Next we matrix multiple Matrix B with its transpose with the following line, we shall call the new 10x10 Matrix Bb.
> Bb <- B %*%b
Excellent, now we have a square matrix.
We again check to see if our Matrix Bb has a determinant. It does!
> det(Bb)
[1] -7.168207e-62
When we try using default solve() on Matrix Bb we had the same issue as with the reciprocal condition number. This time our reciprocal condition number was 2.00809e-22. So we again use the tol= argument.
>solve(Bb,tol = 1e-23)
We get the following result:
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] -5.024464e+08 436218813 -2.181114e+08 5.518944e+08 -6.622717e+07 -2.348820e+08
[2,] -3.046496e+08 167780554 5.032692e+07 1.563031e+08 -1.368673e+08 3.355234e+07
[3,] 4.865526e+08 -369109760 5.033856e+07 -3.523341e+08 1.174416e+08 3.355622e+07
[4,] 2.967027e+08 -268443756 2.684398e+08 -3.249584e+08 2.825681e+07 2.479161e+03
[5,] 1.851943e+12 -74178902 -1.501079e+08 -3.703764e+12 1.851996e+12 2.295835e+08
[6,] 2.825615e+08 -268435510 2.684376e+08 -2.966900e+08 1.412905e+07 -2.856441e+03
[7,] -3.704025e+12 349674560 -3.373140e+08 7.407639e+12 -3.703661e+12 1.059898e+07
[8,] 6.861096e+08 -436217465 2.181122e+08 -6.507888e+08 -1.854142e+07 2.348797e+08
[9,] 1.851632e+12 194264908 -1.501145e+08 -3.703411e+12 1.851685e+12 -3.885154e+07
[10,] -4.944995e+08 268446559 -7.275947e+03 4.521143e+08 4.238336e+07 -2.684369e+08
[,7] [,8] [,9] [,10]
[1,] 2.675599e+08 1.497880e-07 -2.843345e+08 5.032854e+07
[2,] 6.976372e+07 2.684355e+08 -8.653936e+07 -2.181057e+08
[3,] -1.845568e+08 -1.192093e-07 1.677773e+08 5.033446e+07
[4,] -2.826220e+07 -2.684355e+08 2.826075e+07 2.684372e+08
[5,] -1.851946e+12 -1.342177e+08 1.851952e+12 -5.121302e+07
[6,] -2.825631e+08 -8.385764e-08 2.825624e+08 8.892463e+02
[7,] 3.703965e+12 4.111875e-04 -3.703674e+12 -2.666709e+08
[8,] -4.512249e+08 -1.106332e-07 1.995639e+08 2.181072e+08
[9,] -1.851636e+12 1.342177e+08 1.851641e+12 -5.121569e+07
[10,] 2.260583e+08 8.470367e-08 -2.260559e+08 -2.964275e+03
Awesome! Hopefully through this assignment I was able to demonstrate how to manipulate matrices in R using built-in math functions. We also learned how using the help menu in R and looking online we can solve issues and default limitations R may have.
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